Simplicial Structures in Topology

220 VI Homotopy Groups
and
�
y0
H((cy0 ,β ),s)(t)=
β( 2t+s−1
s+1
0 ≤ t ≤ 1−s
2
≤ t ≤ 1.
) 1−s
2
(VI.3.8) Theorem. Let (Y,y0) be a given H-space; then, for every n ≥ 1, the
homotopy group πn(Y,y0) is Abelian.
Proof. Due to the definitions of multiplication in Y and comultiplication in S n ,and
also to [cy0 ] being the identity element of πn(Y,y0), the homotopies
f ∼ σ( f ∨ cy 0 )νn ∼ μY ( f × cy 0 )Δ = f ′
g ∼ σ(cy 0 ∨ g)νn ∼ μY (cy 0 × g)Δ = g ′
hold true for every f ,g ∈ Top ∗(S n ,Y ); therefore,
σ( f ∨ g)νn ∼ σ( f ′ ∨ g ′ )νn ∼ μY ( f ′ × g ′ )Δ
σ(g ∨ f )νn ∼ σ(g ′ ∨ f ′ )νn ∼ μY (g ′ × f ′ )Δ.
However, for every x ∈ S n , either f ′ (x) or g ′ (x) must equal y0 and so, ( f ′ × g ′ )Δ is
a map from S n to Y ∨Y; consequently, we have the homotopies
We end this proof by observing that
μY ( f ′ × g ′ )Δ ∼ σ( f ′ × g ′ )Δ
μY (g ′ × f ′ )Δ ∼ σ(g ′ × f ′ )Δ.
σ( f ′ × g ′ )Δ = σ(g ′ × f ′ )Δ. �
(VI.3.9) Theorem. For every n ≥ 2 and every (Y,y0) ∈ Top ∗, πn(Y,y0) is Abelian.
Proof. It is a consequence of Theorems (VI.3.7), (VI.3.8), and of the fact that ΩY
is an H-space. �
(VI.3.10) Theorem. For every n ≥ 2,
is a covariant functor.
πn : Top ∗ −→ Ab
Proof. A based map k : (Y,y0) −→ (X,x0) induces a group homomorphism
as follows: for every [ f ] ∈ πn(Y,y0),
πn(k): πn(Y,y0) −→ πn(X,x0)
πn(k)([ f ]) :=[kf].
We must use the fact that σ(k ∨ k)=kσ. �